# Bijection iff Left and Right Inverse/Corollary

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## Corollary to Bijection iff Left and Right Inverse

Let $f: S \to T$ and $g: T \to S$ be mappings such that:

\(\ds g \circ f\) | \(=\) | \(\ds I_S\) | ||||||||||||

\(\ds f \circ g\) | \(=\) | \(\ds I_T\) |

Then both $f$ and $g$ are bijections.

## Proof

Suppose we have such mappings $f$ and $g$ with the given properties.

From Bijection iff Left and Right Inverse, we have that $f$ is a bijection, by considering $g = g_1$ and $g = g_2$.

It directly follows that by setting $g = f, f = g_1, f = g_2$, the result Bijection iff Left and Right Inverse can be used the other way about.

$\blacksquare$

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 2$: Product sets, mappings - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 25.1$: Some further results and examples on mappings